I think your wording is the key—coming up with a proof is creating deep understanding, but writing a proof very much need not be explaining or creating deep understanding. Writing a proof can be done mechanically, by both instructor and student, and, if done so, neither demonstrates nor creates understanding.

(Also, in statistics more than in almost any other mathematically based subject, while the rigorous mathematical foundations are important, a complete theoretical understanding of those foundations need not shed any light on the actual practice of statistics.)